The question asks which graph represents two functions, $h(x)$ and $j(x)$, that are decreasing across all points of the domain common to both functions.

Step-by-step Analysis:

1. Key Idea: A function is decreasing in an interval if, as $x$ increases within that interval, the function's value decreases.

2. Common Domain:

   • The graphs show a specific interval where both functions overlap. In that interval, both $h(x)$ and $j(x)$ need to be decreasing for the graph to satisfy the condition.

3. Graph Observations:

   • First graph: $h(x)$ (blue curve) is decreasing in the left half of the domain, but $j(x)$ (orange curve) is increasing, disqualifying this graph.
   • Second graph: $h(x)$ is decreasing where it overlaps with $j(x)$, and $j(x)$ is also decreasing within their common domain.
   • Third graph: $h(x)$ is decreasing, but $j(x)$ is increasing in the common domain.
   • Fourth graph: $h(x)$ is decreasing, but $j(x)$ is increasing in the common domain.

Correct Answer:

The second graph satisfies the condition where both $h(x)$ and $j(x)$ are decreasing across the domain common to both functions.

Would you like a more detailed explanation of how I analyzed the graphs or why specific graphs are disqualified?

Here are 5 related questions for further exploration:

1. What mathematical methods can you use to determine whether a function is decreasing from its equation?
2. How can you find the common domain of two functions algebraically?
3. Can two functions be decreasing over disjoint intervals and still satisfy a similar condition?
4. How does the derivative help determine whether a function is decreasing or increasing?
5. What happens if one function is constant in the common domain—does that count as "decreasing"?

Tip: Remember, a decreasing function always has a negative slope or derivative in the interval of interest!